nLab simplicial bar construction



Homotopy theory

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The simplicial bar construction is a way of building a cofibrant resolution of a diagram, suitable for computing homotopy colimits in simplicial model categories.


Let 𝒱\mathcal{V} be a cocomplete symmetric monoidal closed category and let \mathbb{C} be a small 𝒱\mathcal{V}-category. We define a simplicial object W (c,,c)W_\bullet (c', \mathbb{C}, c) for each pair (c,c)(c', c) of objects in \mathbb{C} by

W n(c,,c)= (c 0,,c n)(c n,c)(c n1,c n)(c 0,c 1)(c,c 0)W_n (c', \mathbb{C}, c) = \coprod_{(c_0, \ldots, c_n)} \mathbb{C} (c_n, c') \otimes \mathbb{C} (c_{n-1}, c_n) \otimes \cdots \otimes \mathbb{C} (c_0, c_1) \otimes \mathbb{C} (c, c_0)

with the obvious face and degeneracy operators induced by the composition and identities of \mathbb{C}. Note that W (blank,,blank)W_\bullet (\blank, \mathbb{C}, \blank) is a 𝒱\mathcal{V}-functor op[Δ op,𝒱]\mathbb{C}^{op} \otimes \mathbb{C} \to [\mathbf{\Delta}^{op}, \mathcal{V}].

Let \mathcal{M} be a tensored 𝒱\mathcal{V}-category. The two-sided simplicial bar construction of a 𝒱\mathcal{V}-diagram F:F : \mathbb{C} \to \mathcal{M} weighted by a 𝒱\mathcal{V}-functor G: op𝒱G : \mathbb{C}^{op} \to \mathcal{V} is a simplicial object B (G,,F)B_\bullet (G, \mathbb{C}, F) equipped with isomorphisms

(B n(G,,F),M) (c,c): op𝒱(W n(c,,c),(GcFc,M))\mathcal{M} (B_n (G, \mathbb{C}, F), M) \cong \int_{(c', c) : \mathbb{C}^{op} \otimes \mathbb{C}} \mathcal{V} (W_n (c', \mathbb{C}, c), \mathcal{M}(G c' \odot F c, M))

that are natural in nn and 𝒱\mathcal{V}-natural in MM, where the integral sign on the right hand side denotes a 𝒱\mathcal{V}-end.

Now suppose a cosimplicial object Δ :Δ𝒱\Delta^\bullet : \mathbf{\Delta} \to \mathcal{V} is given, so that we may define the realisation? of a simplicial object X X_\bullet in a 𝒱\mathcal{V}-category as the weighted colimit |X |=Δ X \left| X_\bullet \right| = \Delta^\bullet \star X_\bullet. The two-sided bar construction of F:F : \mathbb{C} \to \mathcal{M} weighted by G: op𝒱G : \mathbb{C}^{op} \to \mathcal{V} is then defined to be the geometric realisation of the two-sided simplicial bar construction:

B(G,,F)=|B (G,,F)|B (G, \mathbb{C}, F) = \left| B_\bullet (G, \mathbb{C}, F) \right|



If \mathcal{M} is a tensored 𝒱\mathcal{V}-category and has coproducts for small families of objects, then \mathcal{M} has two-sided simplicial bar constructions for all diagrams and weights. If \mathcal{M} is moreover 𝒱\mathcal{V}-cocomplete, then \mathcal{M} also has two-sided bar constructions.


If \mathcal{M} is a tensored 𝒱\mathcal{V}-category and has coproducts for small families of objects, then:

  • For each 𝒱\mathcal{V}-diagram F:F : \mathbb{C} \to \mathcal{M}, the 𝒱\mathcal{V}-functor B n(,,F):[ op,𝒱]B_n (-, \mathbb{C}, F) : [\mathbb{C}^{op}, \mathcal{V}] \to \mathcal{M} has a right 𝒱\mathcal{V}-adjoint.

If \mathcal{M} is also cotensored, then:

  • For each weight G: op𝒱G : \mathbb{C}^{op} \to \mathcal{V}, the 𝒱\mathcal{V}-functor B n(G,,):[,]B_n (G, \mathbb{C}, -) : [\mathbb{C}, \mathcal{M}] \to \mathcal{M} has a right 𝒱\mathcal{V}-adjoint.

Let W(c,,c)W (c', \mathbb{C}, c) be the realisation of W (c,,c)W_\bullet (c', \mathbb{C}, c). If \mathcal{M} is a cotensored 𝒱\mathcal{V}-cocomplete 𝒱\mathcal{V}-category, then there are isomorphisms

(B(G,,F),M) (c,c): op𝒱(W(c,,c),(GcFc,M))\mathcal{M} (B (G, \mathbb{C}, F), M) \cong \int_{(c', c) : \mathbb{C}^{op} \otimes \mathbb{C}} \mathcal{V} (W (c', \mathbb{C}, c), \mathcal{M} (G c' \odot F c, M))

that are 𝒱\mathcal{V}-natural in MM.

The next theorem is essentially a version of the Fubini theorem for coends.


Let F:F : \mathbb{C} \to \mathcal{M}, G:𝔻 op𝒱G : \mathbb{D}^{op} \to \mathcal{V}, and H: op×𝔻𝒱H : \mathbb{C}^{op} \times \mathbb{D} \to \mathcal{V} be 𝒱\mathcal{V}-functors. If \mathcal{M} is a cotensored 𝒱\mathcal{V}-cocomplete 𝒱\mathcal{V}-category, then we have the following isomorphism:

B(B(G,𝔻,H),,F)B(G,𝔻,B(H,,F))B (B (G, \mathbb{D}, H), \mathbb{C}, F) \cong B (G, \mathbb{D}, B (H, \mathbb{C}, F))


As stated in the introduction, the two-sided simplicial bar construction can be used to compute homotopy colimits.

Let us write ycy c for the representable 𝒱\mathcal{V}-functor (,c)\mathbb{C}(-, c) and B(,,F)B (\mathbb{C}, \mathbb{C}, F) for the diagram cB(yc,,F)c \mapsto B (y c, \mathbb{C}, F).


Let \mathcal{M} be a simplicial model category and let \mathbb{C} be a small category.

  1. For each diagram F:F : \mathbb{C} \to \mathcal{M}, there is a weak equivalence B(yc,,F)FcB (y c, \mathbb{C}, F) \to F c that is natural in FF and cc.
  2. The functor colim:[,]colim : [\mathbb{C}, \mathcal{M}] \to \mathcal{M} preserves weak equivalences between diagrams of the form B(,,F)B (\mathbb{C}, \mathbb{C}, F) where FF is a diagram such that each FcF c is a cofibrant object in \mathcal{M}.
  3. The functor B(1,,Q):[,]B (1, \mathbb{C}, Q -) : [\mathbb{C}, \mathcal{M}] \to \mathcal{M}, where Q:Q : \mathcal{M} \to \mathcal{M} is a cofibrant replacement functor, is a left derived functor for colim:[,]colim : [\mathbb{C}, \mathcal{M}] \to \mathcal{M}, i.e. it computes homotopy colimits.


Last revised on August 3, 2013 at 23:56:05. See the history of this page for a list of all contributions to it.